KRF
Knowledge Representation Framework Project
Department of Computer and Information Science, Linköping University,
and Unit for Scientific Information and Learning, KTH, Stockholm
Erik Sandewall
Defeasible Inheritance with Doubt Index
and its Axiomatic Characterization
This is the author’s copy of an article that has been published in the journal Artificial Intelligence in its
December, 2010 issue. The copyright for the article belongs to Elsevier Science.
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2010-07-23
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1
Abstract
This article introduces and uses a representation of defeasible inheritance
networks where links in the network are viewed as propositions, and where
defeasible links are tagged with a quantitative indication of the proportion
of exceptions, called the doubt index. This doubt index is used for restricting
the length of the chains of inference.
The representation also introduces the use of defeater literals that disable
the chaining of subsumption links. The use of defeater literals replaces the
use of negative defeasible inheritance links, expressing “most A are not B”.
The new representation improves the expressivity significantly.
Inference in inheritance networks is defined by a combination of axioms
that constrain the contents of network extensions, a heuristic restriction
that also has that effect, and a nonmonotonic operation of minimizing the
set of defeater literals while retaining consistency.
We introduce an underlying semantics that defines the meaning of literals
in a network, and prove that the axioms are sound with respect to this
semantics. We also discuss the conditions for obtaining completeness.
Traditional concepts, assumptions and issues in research on nonmonotonic
or defeasible inheritance are reviewed in the perspective of this approach.
1
Background and Overview
Nonmonotonic or defeasible inheritance is one of the classical topics in
Knowledge Representation. It concerns structures where there is a number of classes, a subsumption predicate whereby one class can be subsumed
by several superior classes, and a defeasible variant of the subsumption
predicate where the propositions “A is defeasibly subsumed by B” and “B
is defeasibly subsumed by C” allow one to infer “A is defeasibly subsumed
by C” unless there is information to the contrary. Propositions of this kind
are commonly called links; the classes are referred to as nodes in inheritance
networks.
1.1
Earlier Work on Nonmonotonic Inheritance
The path-based approach to this topic defines methods for identifying paths,
i.e. sequences of subsumption predicateships or subsumption-related predicateships in the given inheritance network describing the application at
hand. Paths may be related as e.g. situators, preemptors, conflictors or defeaters, according to their structure, and this is used to define what paths are
permitted by a given inheritance network [TTH91]. Skeptical approaches define one single extension consisting of permitted paths; credulous approaches
define a set of permitted extensions allowing for different possibilities. A
credulous system can either select one of these extensions on extralogical
grounds, or use the intersection of the permitted extensions.
Extensions are usually characterized using rules that define or at least constrain the status of the various paths in a given inheritance network. How-
2
ever, it would be useful to have an axiomatic representation of these constraints in the sense of a set of logic formulas that are satisfied in (permitted)
extensions. This may facilitate the formal analysis of those constraints, and
it should also be a first step towards integrating defeasible-inheritance information with other information about the domain at hand.
Given that defeasible inheritance is generally recognized as an example of
nonmonotonic logic, it will not be sufficient to merely have a set of axioms;
one also needs an appropriate nonmonotonic reasoning policy in the form
of, for example, a circumscription policy or a preference predicate on extensions. We shall use the term “axiomatic representation” for the combination
of a set of axioms and a nonmonotonic reasoning policy.
[San86] proposed one such axiomatic representation and validated it by
applying it to a set of test cases that were widely used in the literature at
the time. [Sim96] observed that this representation obtained unintended
results for some additional test cases that had emerged later, and proposed
a modification of Sandewall’s representation. However, she also observed
some other cases that even the modification did not handle as intended.
[Sch93] has shown that no path-based approach to skeptical reasoning (in
a reasonable sense of that term) can produce the intersection of credulous
extensions. [TTH91] therefore observed that “we cannot even axiomatize
ideal skepticism in our purely path-based formalism”.
Unfortunately, the relation between the scenarios to be represented and the
proposed representation was never obtained in a systematic way in these
works; it was always only shown by way of examples. Both [San86] and
[Sim96] remark that it would be desirable to have an underlying semantics
for defeasible networks, but no semantics-based axiomatic representation
has yet been proposed.
Some of the more recent works on nonmonotonic inheritance pay less attention to the traditional representational questions and focus instead on
technical aspects of the nonmonotonic logic being used, such as its complexity properties, or the introduction of a priority ordering on the links
in the network, which are sometimes represented as default rules [BH95,
NW01, Hor07]. We propose that it is important to check new theories of
defeasible inheritance against a sufficient number of known, difficult cases
of node configurations, and this aspect is emphasized in the present article.
1.2
Overview of the Article
This article consists of the following sections:
1. The present section
2. The investigated approach: the characteristic features of how we represent nonmonotonic inheritance, but without going into technical
details
3. Representation of inheritance networks: defines the formal representation of a network as a set of propositions, and defines the concept
of extension of an inheritance network
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4. Axioms and other restrictions: specifies the restrictions that we propose to apply to network extensions, i.e., the axiom part of the axiomatic representation
5. The Proportion Semantics: the underlying semantics which is used for
verifying the soundness of the axioms in the axiomatic representation,
and for analyzing possible nonmonotonic policies in it
6. Explanation of the axioms
7. Inference operation: definition and motivation for the nonmonotonic
policy component of the approach
8. Issues in commonsense inheritance: relates the approach and the results of the present article to some standard examples and issues in
the area
9. Object-level predicates and description logics: discusses an extension
of the expressivity of the approach that is used here
10. Alternative approaches to defeasible inheritance
11. Conclusion
Appendix 1 contains a discussion of the possibility of also proving completeness for the set of axioms and suggests an approach to doing this. Appendix
2 contains a number of additional examples besides those that are found in
the text.
2
2.1
The Investigated Approach
Characteristic Features of the Representation
In this article we investigate an approach to the representation of defeasible
inheritance that has two characteristic features. First, defeasible subsumption links are annotated with a doubt index, i.e., a number indicating the
extent to which exceptions must be expected. A doubt annotated subsumption link between two classes c and d may be written c subm d, where m
is usually a small integer when given by the sources. These doubt indices
are used to control the transitivity of defeasible subsumption, so that from
c subm d and d subn e one will defaultwise conclude c subm+n e. However, the representation requires the use of a threshold K that sets a limit
to the chaining, so that the conclusion in the example just shown is only
accepted if m + n ≤ K. This provides a cut-off point beyond which further
chaining is not admitted.
The use of doubt annotated links distinguishes our approach from earlier
approaches to defeasible inheritance, and indeed from earlier approaches to
nonmonotonic reasoning in general.
Secondly, the present approach corrects a deficiency in many earlier approaches to inheritance networks which can be illustrated by the following
example. Consider a distinction between “white birds” and “grey birds”.
Most doves are grey. In my park there is a lot of birds; most of them are
doves. However, about half of them are white and the other half are grey.
We therefore have the sub predicate from birds-in-my-park to doves, and
from doves to grey birds, and we wish to override the transitivity that is
4
otherwise obtained by default. This is done using an additional ‘defeater’
predicate nsub in our approach. Most traditional approaches have only one
way of suppressing an inferred subsumption, namely by asserting or inferring another subsumption that is incompatible with the one in question,
and such approaches can not express scenarios such as this one.
The defeater predicate is analogous to exception links which were used in
some of the earliest work on nonmonotonic inheritance [Tou86] but which
have largely fallen into oblivion.
Concurrently with the present work, Gabbay and Schlechta have proposed
an approach to defeasible inheritance using reactive diagrams [?] which also
uses exception links. However, the use of a doubt index does not have any
counterpart in their approach.
In fact, the defeater predicate serves two purposes in the approach that is
studied here. Besides its use for characterizing an application, like in the
example just given, it is also used as the predicate that is to be minimized
for the purpose of obtaining conclusions nonmonotonically, similar to the
use of abnormality predicates in circumscription.
2.2
Motivation for the Proposed Approach
The major reason for the proposed approach is its improved expressivity.
The use of an explicit defeater predicate provides expressivity that the traditional approach using positive and negative, defeasible links does not have,
as illustrated by the grey doves scenario above. Likewise, the use of a doubt
index makes it possible to restrict the number of inference steps, which is
natural since additional uncertainty is introduced in every such step.
Each link in a network is considered as an elementary proposition, and a
network is therefore a set of literals. This makes it possible to specify axioms
and other restrictions that define the inference operation in an inheritance
network. An additional advantage of the use of doubt indices and defeater
literals is that it forms the basis for the definition of a precise underlying
semantics the inheritance networks. This aspect of the approach makes it
possible to investigate the formal properties of the inference system, beginning with a verification of the soundness of the set of axioms.
There are also other representational problems in the traditional approach
that go away with the proposed new approach. For example, with the
traditional approach one obtains circular link structures even in quite simple
situations, like for disjoint or almost-disjoint classes, or for a case where
most A are B but most B are not A, i.e. when A is a small subclass of
B. A semantics for such circular structures has been proposed by Wang et
al [?]. Our approach using a defeater predicate does not obtain a circular
structure in such cases. An assumption of noncircularity in inheritance
networks appears to facilitate proofs and computational procedures, and in
our approach it is an insignificant restriction from the application point of
view.
Last but not least, the underlying semantics provides a precise basis for identifying the expected conclusions in many of the common scenario examples.
It is not satisfactory to define expected conclusions in terms of intuition and
presumed common sense, and the underlying semantics provides a better
basis in this respect.
5
2.3
User Aspects and Underlying Semantics
The use of a doubt index and a defeater predicate provides additional expressivity for the user of the representation. As always, there is a question
of what this requires from the user. On what basis will the user select the
doubt indices on the various inheritance links, and will it be practically
possible to identify the explicitly stated instances of the defeater predicate?
One possible answer to these questions is to present the nonmonotonic reasoning system with its notation and its inference mechanism as a kind of
machine that it is up to the user to use and to master. For example, the
user may be advised to start by using the doubt index 1 on each defeasible inheritance link, so that the threshold K simply specifies the maximum
permissible number of links in subsumption paths, and then to use other
values for the doubt index if there is a particular reason for doing so. One
leaves it to the user to select doubt indices and to introduce defeater literals
until her representation of her application has been sufficiently debugged in
the sense of providing the intended results when tested.
Although this “practical engineering” approach may be the only possible
one in practical situations, it is obviously not satisfactory from a scientific
point of view, nor in the long run even for practical purposes.
First of all, the defeasible inheritance network that is provided as the input
to the inference process should be viewed as a domain model of the application at hand. This means in particular that inasmuch as default conclusions
are those that are drawn in the absence of contrary information, the application model shall be assumed to be epistemologically complete with respect to
information about exceptions – meaning that it is as complete as is required
for the purpose at hand. If the application model omits significant information about exceptions, then it is not possible for an inference mechanism
to compensate for it. Furthermore, if an epistemological incompleteness is
detected, then it should be repaired by adding information that characterizes the exception in question, and not by tinkering with the model in some
other way.
This principle requires that the representational conventions that are used
for the domain model must be able to express exception information, and
that the semantics of such information must be well defined. In the approach that was outlined in Section 2.1, in particular, we need definitions
of what are the appropriate values for the doubt indices and what are the
appropriate choices of the defeater predicate, based on known properties of
the application at hand. We shall use the term underlying semantics for
such a formal basis.
Notice that an underlying semantics can be useful even if there is no practical way of calculating exact doubt index values in a given situtation. It
may be that one can define a learning process where these values are adjusted continuously based on experience, and the underlying semantics is
relevant for defining the learning process. It may also be that instances of
the defeater predicate that have been inferred in some reasoning tasks, can
then be accepted to the knowledge base in their own right and for use at
later times.
Furthermore, even in those circumstances where the “practical engineering”
approach is in fact adopted, one may consider the possibility of using the
6
definitions of appropriate values and choices as heuristic guidelines when the
aforementioned nonmonotonic reasoning “engine” is to be equipped with
contents.
Last but not least, as one proceeds to defining rules of inference or other
inference machinery for the representation, it is appropriate to have a definition of the meaning of index values and of defeater literals as a basis. One
may of course object that this does not matter in the case of the “practical
engineering” approach, since in its case the formalism and the machinery
are presented to the user and it is up to her to make the best use of it,
regardless of whether its design is systematic or ad hoc, but the existence of
an underlying semantics ought to be reassuring for designer and user alike.
2.4
Approach for the Underlying Semantics
Our representation of inheritance networks uses uniformly one type of things,
called classes. Each class is supposed to have a finite and non-empty set
of members. These members are objects, but objects are only used in the
underlying semantics and not in the representation system. In addition the
representation system uses the domain of the non-negative real numbers as
values of the doubt index. We use the term ‘class’ alternatingly for the
abstract entity and for the symbol representing that entity.
Although the representation system does not have any constructs for objects
per se, we shall distinguish between two kinds of classes, namely singleton
classes that have exactly one member each, and concept classes that have
an arbitrary number of members, and normally a considerable number of
them.
The basic idea in the underlying semantics is to interpret a doubt annotated
subsumption link c subm d as a statement “the proportion of members of
c that are members of d is at least p” where p is a number slightly smaller
than 1, or equal to 1, and where there is a one-to-one mapping between the
doubt index m and the proportion p. This is the same as saying that the
probability that a randomly drawn member of c shall be a member of d, is
at least p.
Notice that the interpretation is that the proportion in question is at least
p and not that it is p. This is because there may be more than one chain of
links from c to d in the inheritance network at hand which result in different
values for p. If the interpretation had been that the proportion in question
is p then one would obtain an inconsistency in such cases.
If the proportion of c in d is at least p and the proportion of d in e is at
least q, then the proportion of c in e is at least p × q unless the distribution
of the c’s in d is uneven in the direction of the part of d that is not in e.
The index value m in c subm d must therefore be interpreted essentially as
the logarithm of p, so that the multiplication of proportions is mapped to
the addition of their logarithms. More exactly, however, m is interpreted as
−Φ × e log(p) where Φ is a constant whose purpose is to obtain values that
are often in the range of 1 to 10, or 1 to 100, rather than small fractions
of 1. Notice that if p = 1 − ε for a small ε then e log(p) is approximately
−ε. The minus sign in the interpretation of m obtains that doubt index
values are positive rather than negative. The function ϕ is defined so that
the relation between m and p will be written as p = ϕ(m).
7
The caveat that the multiplication of proportion values only applies if the
distribution of the c’s in d permits it is an essential precondition for the
formal analysis. It must also be respected when the approach is used for a
practical application, to the extent that it is practically possible to respect
it. The defeater predicate is the manifestation of that precondition in the
axiomatic representation.
2.5
Singleton Classes
The interpretation of the doubt index described above applies to the case
where the nodes in the inheritance network represent sets with a certain
number of members (concept classes), and it is not applicable for nodes
that represent a single object (singleton classes).
The underlying semantics is defined only for concept classes, but actual
uses of inheritance networks must accomodate both singleton and concept
classes. In fact, one important type of query for an inheritance network has
the form c subm d where c is a singleton class and d is a concept class. It
is therefore important to provide an interpretation of such queries as well.
We distinguish the following two cases. In a simple case, all links that
involve singleton classes in the given network are strict links that do not
admit any doubt, i.e. links of the form c sub0 d where c is a singleton
class and d is a concept class. The inference process may identify inheritance
chains resulting in conclusions of the form c subm e where m > 0. The
interpretation for the user of such a conclusion will then be: “there is some
class d in the network that contains the sole member of c such that the
probability of a random member of d being a member of e, is at least
ϕ(m).” From the application point of view this may be abbreviated as “the
member of c is a member of d with probability ϕ(m).”
In the more general case links of the form c subk d with singleton c and
a nonzero value of k are admitted in the given inheritance network as well.
The underlying semantics does not provide any interpretation for such a
link. From an application point of view it is natural to consider it as an
expression of a probability, viz., that the probability of c being in d is
at least ϕ(k), but the nature of that probability is outside the scope of
the underlying semantics. We shall refer to it as an external probability.
In this case, the interpretation for the user of a conclusion of the form
c subm e will be: “there is some class d in the network such that the
probability is at least ϕ(k) that the sole member of c is a member of d, and
the probability of a random member of d being a member of e, is at least
q, where ϕ(k) × q = ϕ(m).”
From the application point of view it may well be appropriate to express
this more concisely by saying that the probability of c being in e is at least
ϕ(m), but from a formal point of view this combines probability information
from two different sources with different character.
Links of the form c subm d where both c and d are singleton classes must
also be interpreted in terms of external probabilities, to the extent that such
links are allowed to occur.
Therefore, given sub links with a singleton first argument are not to be used
in the defeasible inference process, but shall only be taken as statements of
an externally provided probability. Since they have no essential relevance for
8
the topic of the present article, we shall make the simplifying assumption
that given inheritance networks only contain strict subsumption links for
singleton classes.
3
Representation of Inheritance Networks
The representation system is the entire set of conventions, assumptions and
restrictions that are used for expressing domain information and for making
inferences from it. The conventions for representing an inheritance network
as a set of formulas is the first part of the representation system.
3.1
Predicates and Inheritance Networks
The representation system is based on named classes each of which has a
set of members, which are called objects. Applications may associate other
information with classes, besides their sets of members, and it is therefore
possible to have two classes that have equal sets of members but which are
anyway different classes. However, from the formal point of view and in
this article, it is only the set of members that is of interest. Therefore, all
the predicates on classes are defined in terms of their member sets.
Five predicates on classes are used, namely v, dj, sub, dsub and nsub, with
the following intended meanings:
• a v b means that the members of a is a subset of or equal to the
members of b
• a dj b means that the classes a and b do not have any common
member
• a subm b means that according to given sources, most members of
a are members of b, where m is the doubt value, expressed as a real
number
• a dsubm b means that it has been inferred that most members of a
are members of b, with m again being the doubt value
• nsub(a,b,c) is used to represent a defeater literal that suppresses
the default of chaining a dsub b and b sub c.
We shall return to the distinction between sub and dsub in subsection 4.5.
An expression consisting of one of these predicates with its proper arguments
will be called a literal. This corresponds to a “link” in many earlier articles.
If the classes that occur as arguments in the literal belong to a set C of
classes, then it is a literal over C.
In order to avoid having to define inference steps that are trivial conclusions
in set theory for the sets of members for classes, we introduce an assumption to the effect that these trivial conclusions are already present in given
inheritance networks. The following definition is used. Let S be a set of
singleton classes and C be a set of concept classes. A network kernel over
S and C is a set of literals over S ∪ C using the v and dj predicates that
is satisfiable and saturated inferentially, so that it contains all possible conclusions with respect to those two predicates. Furthermore the kernel must
9
contain literals of the form c dj d for every pair of different singleton class
symbols, which constitutes a unique names assumption.
The satisfiability requirement means that the kernel can not contain e.g. a
literal of the form c dj c. (Recall that each class must have a nonempty
set of members). It also can not contain a literal of the form c v d where
c and d are different singleton classes.
An inheritance network is a fivetuple hS, C, Γ, ∆, Λi where S is a set of
singleton classes, C is a set of concept classes, Γ is a network kernel over
S and C, ∆ is a set of literals over C alone using the sub, dsub and nsub
predicates, and Λ is a set of literals using the sub, dsub and nsub predicates
where the first argument is in S and the other argument(s) are in C.
Notice that only positive literals occur in inheritance networks.
Unlike most earlier approaches, in this system there is no predicate for
“members of a are usually not members of b”. Such situations have to
be expressed in some other way, for example using an additional class c
whereby one can write a subm c and c dj b.
3.2
Given Inheritance Networks and their Extensions
The following conventions and definitions will be used in defining how to
make inference from inheritance networks.
The set of those literals in an inheritance network that use a particular
predicate R will be called the contents for R of the inheritance network.
Consider two inheritance networks N = hS, C, Γ, ∆, Λi and N 0 = hS, C, Γ, ∆0 , Λ0 i
which means that they have equal kernels. Then N 0 is said to be an extension of N iff ∆ ⊆ ∆0 and Λ ⊆ Λ0 .
Like in most other work on inheritance networks, the inference operation is
defined as an operation that takes an inheritance network as input, called the
given network, and that produces an extension of that network, containing
conclusions that can reasonably be drawn from the given network. This
extension will be called the derived extension.
We make the following particular assumptions about the given inheritance
network N = hS, C, Γ, ∆, Λi in the inference operation: the ∆ component
may only use the predicates sub and nsub, but not dsub, and the Λ component must only use the nsub predicate. This is because dsub is to be
used for conclusions, and not for originally given links. The restriction for
Λ expresses that singleton classes are only allowed to participate in strict
links such as c v d, and not in defeasible links. Notice however that these
restrictions only apply for the given inheritance networks, and not for its
derived extensions.
This framework imposes a strong restriction on the expressivity of conclusions, compared to what is the case in logic in general. Single literals can
be obtained as conclusions; conjunctions of literals can be obtained since
the literals in a network are implicitly conjunct. Negation of a proposed literal a v c can be obtained indirectly if the extension contains a v b and
b dj c for some b, since each class is assumed to have a nonempty set of
members. Negations of sub, dsub and dj can be obtained in similar ways.
However, a complement set can not be represented, nor can a disjunction of
10
literals be obtained as a conclusion according to this view of inference, and
the same applies of course for a negation of conjuncts. (A generalization in
these respects will be briefly described in Section 9).
4
Axioms and Other Restrictions
We proceed now to the operation that obtains the derived extension of a
given network.
4.1
Notation for Extension Constraints
The most natural way of defining the derived extension of a given inheritance
network may seem to be to introduce a set of inference rules, and to define
the derived extension as the set of all literals that can be obtained from the
given network by successive use of these inference rules. However, we choose
instead to define the derived extension using extension constraints, that is,
a set of logic formulas that impose restrictions on the permissible contents
and structure of extensions, since this provides a natural way of expressing
and analyzing a relatively complex inference mechanism. The axioms of the
axiomatic representation are introduced as extension constraints.
The notation for expressing extension constraints is separate from, and
richer than the notation for the inheritance networks themselves, and is
defined as follows. We assume a set of variable symbols that is disjoint
from the set of identifiers for classes, and which is partitioned into a set of
class variables and a set of numerical variables. No type distinction is made
between variables for singleton classes and those for concept classes. A numerical term is defined recursively as a number, a numerical variable, the
symbol K, or a composite expression that is formed from numerical terms
using the addition function.
An atomic proposition is either an expression x ≤ y where x and y are
numerical terms, or an expression that is similar to a literal but with class
variables instead of class identifiers, and with numerical terms instead of
numbers. A proposition is formed recursively from atomic propositions using the standard propositional connectives, in particular ∧, ¬, → and ↔.
Quantifiers are not used.
Extension constraints will be expressed as propositions.
The evaluation of a proposition can now be defined. Let the following be
given: an inheritance network hS, C, Γ, ∆, Λi; sets of variables for classes
and for numbers; mappings from class variables to members of S ∪ C and
from numerical variables to non-negative real numbers; and finally a specific
positive number called K. The value of a variable is always the class identifier or the number assigned to it by the respective variable-value mappings.
The value of K is K, and the value of a number is itself. The value of a
numerical term of the form m + n is obtained as the sum of the values of
the sub-expressions m and n.
The truth-value of an atomic proposition is obtained as follows. If the
proposition has the form x ≤ y then it is T if the value of x is less than or
equal to the value of y, and F otherwise. For all predicates having classes as
11
arguments, substitute the arguments containing class variables with their
value according to the mapping for class variables. Also, if the predicate
in the proposition is sub or dsub then substitute similarly the numerical
term used for the doubt value. If the resulting literal (variable-free atomic
proposition) is a member of Γ ∪ ∆ ∪ Λ then the truth-value is T, otherwise
F.
The truth-value of a composite proposition is obtained using the standard
truth-tables.
Finally, a proposition is satisfied in an inheritance network and for a particular value of K iff its value is T for all possible choices of the mappings
for class variables and numerical variables.
4.2
Axioms and Additional Restriction
The following set of eight axioms will be used for obtaining derived extensions.
1.
2.
3.
4.
5.
6.
7:1.
8.
c v d ↔ c sub0 d
c subm d ∧ d v e → c subm e
c subm d → c dsubm d
¬(c dsubm d ∧ c dsubn e ∧ d dj e)
¬nsub(c,c,d)
c dsubm d → m ≤ K
c dsubm e ∧ e subn g ∧ ¬nsub(c,e,g) ∧ m+n ≤ K
→ c dsubm+n g
c dsubm d ∧ m ≤ n ∧ n ≤ K → c dsubn d
All of these except Axioms 4 and 5 can be read informally as if-then rules
that allow one to infer additional literals from given ones. Under this reading it is also possible to construct quasiformal proofs of derived literals.
However, in strict terms we shall only use these formulas as restrictions on
sets of literals, i.e., on inheritance networks.
Axiom 7:1 is the first member of a sequence of successively stronger axioms
that will be called 7:2, 7:3, etc., where we have, in particular,
7:2. c dsubk d ∧ d subm e ∧ e subn g ∧
¬nsub(c,d,g) ∧ ¬nsub(d,e,g) ∧ k+m+n ≤ K
→ c dsubk+m+n g
Each axiom in that sequence entails its predecessor. We shall return to the
later variants of this axiom when considering the completeness of the set of
axioms. Axiom 7:1 is easy to understand and it suffices in many situations.
In addition to these axioms we shall also use an additional restriction, called
Restriction 9, which is
9. c dsubm d ∧ nsub(d,e,g) → nsub(c,e,g)
It will be just called a restriction, and not an axiom, since its motivation in
terms of the semantics is different from that of the axioms.
The constant K is used since the reliability of the conclusions decreases when
several defeasible links are combined by way of transitivity. The condition
on m + n in Axiom 7:1 is a way of stopping the conclusions when the doubt
value has increased to a certain point. The choice of an appropriate value
for K depends on the needs of the application and on the metric that is
12
used for the assignment of doubt values. The permitted range of values for
K will be specified and explained in Section 5.1. The following definitions
are made under the assumption that a value for K has been fixed.
Notice that Axiom 5 does not force negated literals of the form ¬nsub(c,c,d)
to be included in inheritance networks; it merely prevents literals of the
form nsub(c,c,d) from being included there. Inheritance networks consist
of positive literals only.
Notice also that the combined occurrence of nsub(c,d,e) and c dsubm e
is not a contradiction and may be meaningfully used. Consider for example
the following literals:
c dsub1 d
d dsub1 e
nsub(c,d,e)
c dsub4 e
Here, the occurrence of the nsub literal precludes the conclusion c dsub2 e
that would otherwise have been possible using Axiom 7:1. There is still a
link from c to e, but with a higher doubt index.
4.3
Correct and Valid Inheritance Networks
Definition. An inheritance network is said to be correct iff Axioms 1 to 6
are satisfied in it.
Definition. An inheritance network is said to be valid iff Axioms 1 to 8
are satisfied in it.
Proposition 1. Every correct inheritance network has at least one valid
extension.
Proof. The extension obtained by adding literals nsub(a,b,c) for all combinations of concept classes a, b, c that occur in the given network and
where a is different from b satisfies Axioms 1 to 8.
Definition. An inheritance network is consistent iff it has a valid extension
with the same contents for nsub.
Definitions and results in the sequel will apply to correct inheritance networks, so the role of Axioms 1 to 6 is to enforce wellformedness conditions
on the inheritance networks being considered. For example, a correct inheritance network can not contain inconsistencies such as
c subm d
c subn e
d dj e
with m ≤ K and n ≤ K.
The correctness requirement precludes making those statements with m and
n greater than K. This condition is imposed since otherwise Axiom 4 could
not be maintained.
The effect of Axiom 8 is that in every valid inheritance network, the possible
values for m in c subm d is either the empty set, or a closed interval from
some k to K. In practice it is the left endpoint k of that interval that
one is interested in, and that is also how it should be represented in an
implementation, but for the formal analysis it is technically simpler to allow
the entire interval.
13
4.4
Minimal Extensions of Inheritance Networks
Definition. A valid extension N’ of a given inheritance network N is said
to be dsub-minimal iff there does not exist any other valid extension N” of
N such that N’ is an extension of N”, and N’ and N” have the same contents
for nsub.
Proposition 2. Every correct inheritance network has at most one dsubminimal extension for each choice of the contents for nsub.
The proof is straightforward from the definitions.
It follows that a dsub-minimal extension contains all conclusions from the
given inheritance network that are enforced by the axioms, for the given
choice of nsub, and no others. There is a separate question of nsubminimality where the nsub predicate is minimized, but this is a topic for a
later section.
Definition. The minimal extension of a consistent inheritance network N
is the dsub-minimal extension having the same contents for nsub as N has.
We must also be able to relate extensions with different contents for nsub.
Consider two valid extensions N1 and N2 of a given, correct inheritance
network. The meet of N1 and N2 is also an extension of the given inheritance
network whose contents for nsub is the union of those contents for N1 and
N2 , and whose contents for the other predicates is the intersection of those
contents for N1 and N2 .
Notice in particular the role played by Axiom 8 when this definition is
applied: the intersection of the contents for c dsubm d for given c and d will
obtain the weakest one of the contributions from the two given extensions.
We shall use
Proposition 3. The meet of two valid extensions of a correct inheritance
network is valid.
The proof follows easily by inspection of the eight axioms.
Is the meet of dsub-minimal extensions also dsub-minimal
4.5
Distinction between the Predicates sub and dsub
The distinction between the predicates sub and dsub may at first seem
redundant. Could we just identify those two, and dispense with Axiom 3?
The reason for the distinction between sub and dsub is technical. Let us
first describe it from the point of view of a sequence of inference steps.
Suppose the following information is given from the sources (doubt values
omitted)
a sub b
b sub c
c sub d
d sub e
Both Axioms 1 to 8 and the simplified set of axioms will obtain that a dsub
e is a conclusion, given that sub and dsub are synonyms in the simplified
case. However, if Axioms 1 to 8 are used then the following is the only way
of obtaining the intermediate steps towards the conclusion:
14
a dsub b
a dsub c
a dsub d
a dsub e
whereas with the simplified set of axioms there are several different such
sequences, which may be thought of as proofs. The use of Axioms 1 to 8
obtains the same effect as the ascending construction of inheritance paths
in path-based approaches.
Therefore, if one considers adding a defeater literal in order to block this
conclusion, then one defeater is sufficient if Axioms 1 to 8 are used, for
example nsub(b,c,d). In the simplified alternative one must add several
defeaters, in order to stop all the possible ways of obtaining the conclusion.
The use of the distinction between the sub and dsub predicates has two
advantages, therefore. It is one way (of several possible ways) of avoiding
the combinatorial explosion when chaining along long paths, in those cases
where such chaining is needed. Furthermore, and more importantly, it contributes to keeping down the number and the size of the defeater sets that
must be assumed in order to obtain the intended default conclusions.
5
The Proportion Semantics
We now proceed to the definition of a formal framework for analyzing and
motivating the axioms that were introduced in Section 4.2. We shall define
an underlying semantics which defines whether a literal in an inheritance
network holds in an underlying structure. This underlying semantics will be
used as the basis for studying the properties of the axiomatic representation.
5.1
Underlying structures
We have already defined one semantical level where propositions are evaluated in an inheritance network consisting of literals. The underlying semantics is a lower level where literals in an inheritance network are evaluated in
a more detailed structure and in a nontrivial way. In order not to confound
the two evaluation levels, we shall use the terminology that a literal holds in
the underlying structure, whereas a proposition is satisfied in an inheritance
network according to earlier definitions. The term “the truth of” will be
used in both cases.
An underlying structure is a fivetuple consisting of a nonempty finite object
domain O, a nonempty finite set C of class names, a mapping M that assigns
a nonempty subset of O to each class name, a doubt scale factor Φ which
shall be a large positive number, and a threshold K that is a positive number
< −Φ ∗ e log(0.5). For example, of Φ = 1000 then K must technically be
< 693.14..., although in practice it should be much smaller. The constant
K is like before the quantity that is used as the value of the constant K in
propositions. Normally the domain O is a large set.
If U is an underlying structure then we write KU for the threshold component of U .
15
If A and B are subsets of O and |A| is the cardinality of the set A then we
define prop(A,B) as |A ∩ B| / |B|. For example, if B has 10 members, and
8 of the 10 members of B are in A, then prop(A,B) = 0.8. The number of
members in A is not relevant for this. Therefore prop(A,B) can be thought
of as the conditional probability P(A|B).
Now consider an underlying structure hO, C, M, Φ, Ki, an inheritance network hS, C, Γ, ∆, Λi, and a literal over C. This literal is said to hold in the
underlying structure according to the following conditions:
• c v d holds iff M(c) ⊆ M(d)
• c dj d holds iff M(c) ∩ M(d) is the empty set
• c subm d holds iff
K ≥ m ≥ −Φ ∗ e log(prop(M(d),M(c)))
• c dsubm d holds iff c subm d holds
• nsub(c,d,e) does not hold iff
prop(M(e), M(c) ∩ M(d)) ≥ prop(M(e),M(d))
For example, if M(c) ⊆ M(d) then prop(M(d),M(c)) = 1 and c sub0 d
holds since log(1) = 0. If a few members of a not-so-small c are not in d
then prop(M(d),M(c)) decreases to a number 1 − ² for a small positive ².
The corresponding doubt factor is then approximatively Φ∗² since e log(1−²)
is approximately −² for small ².
Notice that the singleton part of an inheritance network is not considered
in this definition, for the reasons that were discussed in Section 2.5.
A model for an inheritance network is an underlying structure where all the
network’s literals hold.
The reference network for a given underlying structure with C as its set
of class names is the inheritance network consisting of C and the set of
all literals over C that hold in that structure. The set of singleton classes
shall be empty. It follows that any underlying structure is a model for its
reference network.
It follows immediately:
Proposition 4. Let N = hS, C, Γ, ∆, Λi be an inheritance network, let M
be a model for N , and let N ∗ = h∅, C, Γ∗ , ∆∗ , ∅i be the reference network
of M . Then Γ ⊆ Γ∗ and ∆ ⊆ ∆∗ .
These definitions disregard the singleton part of the inheritance network.
The generalization to including singleton classes is straightforward, but
omitted here since it does not affect the results in this article. The notation is prepared for future additional work.
5.2
Soundness of the Axioms
In order to analyze the soundness of the restrictions, we extend the use
of underlying structures from literals to propositions, using the reference
16
network. A proposition P is said to be satisfied in an underlying structure
U iff it is satisfied for KU in the reference network for that structure.
Proposition 5. The Axioms 1 through 8 are satisfied in all underlying
structures.
Proof. For a given underlying structure, consider the set of all literals that
hold there, and verify each of the axioms by considering its value over that
set. The proof is trivial or next to trivial for all axioms except for Axiom 7.
With respect to Axiom 4, the conclusion follows according to the condition
on the value of K. Falsifying this restriction would require that d and e,
which are disjoint sets, both are assigned more than half of the members of
c in the underlying structure.
The validation for Axiom 7 depends on the definition for nsub of holds
and can be characterized as an assumption of at least equal proportion.
Used when c subm d and d subn e, the absence of nsub(c,d,e) is the
assumption that the members of M(c) ∩ M(d) are divided between M(e) and
its complement with at least the same proportion in M(e) as the members
of M(d) are.
We first show the proof with respect to Axiom 7:2. Recall that this axiom
is as follows:
7:2. c dsubk d ∧ d subm e ∧ e subn g ∧
¬nsub(c,d,g) ∧ ¬nsub(d,e,g) ∧ k+m+n ≤ K
→ c dsubk+m+n g
Consider an assignment of values to the variables in this formula where the
antecedents of Axiom 7:2 are satisfied so that
c dsubk d
d subm e
e subn g
k+m+n≤K
and so that nsub(c,d,g) and nsub(d,e,g) are not literals in the given
inheritance network. We then have
K ≥ k ≥ −Φ ∗ e log(prop(M(d),M(c)))
K ≥ m ≥ −Φ ∗ e log(prop(M(e),M(d)))
K ≥ n ≥ −Φ ∗ e log(prop(M(g),M(e)))
Writing C for M(c) and similarly for D, E and G we obtain at once, and since
k > 0,
K ≥ m + n ≥ −Φ ∗ e log(prop(E,D) ∗ prop(G,E))
However, since nsub(d,e,g) is not in the given reference network we also
have prop(G, D ∩ E) ≥ prop(G,E), and we obtain
prop(E,D) * prop(G,E) ≤ prop(E,D) * prop(G, D ∩ E) =
|D ∩ E|/|D| * |D ∩ E ∩ G|/|D ∩ E| = |D ∩ E ∩ G|/|D| ≤
|D ∩ G|/|D| = prop(G,D)
so that
K ≥ m + n ≥ −Φ ∗ e log(prop(M(g),M(d)))
An analogous argument is used for combining this conclusion with
K ≥ k ≥ −Φ ∗ e log(prop(M(d),M(c)))
which we observed above, obtaining
K ≥ k + m + n ≥ −Φ ∗ e log(prop(M(g),M(c)))
In the case of Axiom 7:2 it is therefore required to repeat the same argument
two times. The proof for axiom variants 7:n for larger values of n is entirely
analogous. This concludes the proof.
Proposition 6 (Soundness). If N is a consistent inheritance network
17
then all literals in the minimal extension of N hold in all the models of N .
Proof. This follows immediately from Proposition 5.
Appendix 1 discusses the possibility of a completeness proof. In this context
it also discusses properties and usefulness of circular subsumption structures.
Singleton Classes Revisited - write about this.
6
Effects of Restriction 9 and Axiom 7
We have defined Axiom 7:1 and a stronger variant of it, called Axiom 7:2.
Axiom 7:1 can be obtained by selecting c = d in Axiom 7:2 and using axioms 1 and 5. The need for the stronger variant can be understood through
the following example which first shows the reasons for using Restriction 9.
Consider the following example:
C v RE
RE subk E
E subm GA
nsub(RE,E,GA)
Without Restriction 9 one can conclude C dsubk E and C dsubk+m GA.
This is not a desirable conclusion, since the nsub literal expresses that the
other literals do not allow us to conclude that a randomly chosen member
c of RE is a member of GA with probability ϕ(k + m). Therefore, and in
the absence of additional information about C, we ought not to be able to
conclude C dsubk+m GA. Restriction 9 has the desirable effect of forcing
nsub(C,E,GA) to be added to the accepted extension, which prevents the
conclusion of C dsubk+m GA. This is in line with the standard view in the
literature on defeasible inheritance.
The above example is a well-known schema in the literature on defeasible
inheritance, and is known as “Clyde, the Royal Elephant,” and it is easy
to find other, similar examples. However, one must also consider what is
the effect of Restriction 9 if there is other information that will tend to
override, directly or by way of inference, the conclusion that is obtained
from Restriction 9.
In the direct case there is actually no problem. If C subn GA is added to
the given inheritance network, then there is anyway no contradiction due to
Restriction 9. This is because the occurrence of a literal with nsub does not
contradict the corresponding literal with dsub, it merely prevents it from
being inferred in a particular way.
A slightly more complex example is obtained by adding the following literals
to the original ones.
C v CE
CE subj GA
The conclusion C subj GA is obtained in this example as well. The literal
for nsub does not block it since its middle argument provides the required
selectivity.
The situation is different if the direct link from CE to GA is replaced by a
link from CE to E, as follows:
18
C v CE
CE subj E
The conclusion CE dsubj+m GA follows in this case; it is not affected by
the given and inferred nsub literals. The interesting question is whether C
dsubj+m GA should also be inferred. In the framework of the Proportion
Semantics it is natural to discuss this in terms of probabilities. The example
provides two parallel paths from C to E, namely, through CE and through
RE. If the latter two classes almost coincide then one should not be able to
infer C dsubj+m GA. However, in this case one should also not be able to
infer CE dsubj+m GA, so the given network ought to include the literal
nsub(CE,E,GA)
in order to be a correct representation of the application at hand, in accordance with the requirements on the domain model that were specified
in Section 2.3. If it does not do so, and it does not allow that literal to
be inferred using Restriction 9, then it should come as no surprise that
unwarranted conclusions are obtained.
On the other hand, if the contents of CE are unrelated to the contents of RE,
at least with respect to inclusion in GA, and more specifically if CE dsubj+m
GA is a warranted conclusion, then C dsubj+m GA ought to be so as well.
This is the point where Axiom 7:2 is needed. Axiom 7:1 is sufficiently strong
in very many of the situations where chaining of links is required, but the
present case is an example of where it is not sufficient: Restriction 9 forces
nsub(C,E,GA) to be included in the extension, and this blocks the inference
of C dsub GA both through RE and through CE using Axiom 7:1. However,
Axiom 7:2 is able to bypass the restriction of nsub(C,E,GA) and to allow
inferring the link from C to GA via CE.
We shall use the term off-path preclusion for a situation in an inheritance
network where there is a sub path through some classes c, d, e and g and
also a literal nsx(d’,e,g) where d’ is different from d. The nsx literal
may preclude the use of the sub path, and the node d’ is off-path. (Please
notice that although somewhat related, this is not the distinction between
on-path and off-path preemption systems).
Unfortunately, Axiom 7:2 is not sufficiently strong for all situations; it fails
in examples involving double off-path preclusion, as in the following example.
C v RE
RE subk E
E subm GA
nsub(RE,E,GA)
C v CE
CE subj E
GA subn LA
CE subp CA
CA subq GA
nsub(CA,GA,LA)
The first six literals are the same as above, and the following five literals
adds another, similar off-path preclusion. There is an upward sub chain
from C via CE, E and GA to LA (which could be interpreted as “land animal”,
for example), and the two successive nodes E and GA in that chain occur
as the middle argument of nsub, which blocks the chaining of dsub at that
point. We have seen how Axiom 7:2 allows the inference system to pass by
one such block, but it is not sufficient for passing two successive blocks in a
subsumption chain.
19
At the same time, using the same argument as above, it is reasonable to
expect the system to obtain C subj+m+n LA as a conclusion, since there
is no indication of CE being related to RE, or of CA being related to E. It
is easily seen that Axiom 7:3, constructed by analogy with Axiom 7:2 but
with a chain of three sub literals in its antecedent instead of two, is able to
obtain this conclusion.
By extrapolation, it would be appropriate to use the variant Axiom 7:n
where n is the largest number of consecutive off-path preclusions that may
occur in the inheritance networks being considered. If these networks are
known to be cycle-free for sub and dsub literals then the maximum length
of a chain of such literals will be sufficient, but it seems likely that a significantly smaller value of n will be sufficient in practice. Some empirical
information on the frequency of multi-off-path preclusion situations in actual, large knowledgebases would therefore be of interest.
7
Inference Operation
7.1
Inference of the Defeater predicate
The analysis of Axioms 1 to 8 in Section 5 depended on an assumption that
all applicable literals for the nsub predicate are alredy present in the given
inheritance network. Section 6 discussed the use of Restriction 9 which
allows some literals for nsub to be inferred from others, thereby relieving
the user from the obligation to write out all occurrences of nsub explicitly.
Restriction 9 has a heuristic and cautionary character: it is heuristic in the
sense of not being formally sound according to the underlying semantics,
and it is cautionary in the sense that it tends to suppress default conclusions
when there is a good reason to suspect that the preconditions for drawing
the conclusion are not present.
Traditional nonmonotonic reasoning uses another condition for inferring
instances of a particular predicate, namely, in order to avoid inconsistencies.
If a given set of propositions is inconsistent, then the inconsistency may be
removed by adding literals for an abnormality predicate, assuming of course
that the entire system is set up in such a way that adding those literals will
suppress certain previously obtained conclusions. It is customary to add a
minimal set of such literals under the requirement to add sufficiently many
so that the inconsistency goes away. In our case the nsub predicate can
serve as such an abnormality predicate, and addition of literals for nsub
may turn an inconsistent inheritance network into a consistent one.
However, although removing inconsistency is essential from the point of view
of classical logic, it does not in itself represent a need in the application. We
propose that it is more constructive to think of the addition of abnormality
literals as a critical review process, where the initially given information
(the given inheritance network, in our case) is checked for indications of
anomalies, and abnormality literals are added in a cautionary fashion in
order to avoid making conclusions on dubious grounds.
Consider for example the case of directly conflicting subsumers which is more
often known as the “Nixon Diamond”. Using our notation it is:
N sub Q
20
N
Q
R
P
sub R
sub P
sub NP
dj NP
The annotation of doubt values is not of interest when considering this
example and it has therefore been omitted. It is assumed that doubt values
have been selected in such a way that the restriction using the constant K
is not violated. The same will apply for many of the examples in the sequel.
In order for Axioms 3 and (any variant of) Axiom 7 to be satisfied, unless
there are additional literals for nsub, a valid extension must contain both
of
N dsub P
N dsub NP
which violates Axiom 4. This can only be avoided by adding one or both of
the following
nsub(N,Q,P)
nsub(N,R,NP)
Both of these are equally possible because of the symmetry. Neither Restriction 9 nor any of the axioms will force either of these literals to be added,
so some other mechanism is needed in order to avoid an inconsistency in
this example. The traditional approach is to allow two admitted extensions,
one containing only nsub(N,Q,P), the other containing only nsub(N,R,NP),
with the understanding that one or the other of these must be the case, but
we do not know which.
The problem with this is the following. Assume first that all the classes
in this example are concept classes. Since it has been stated that typical
members of N are members of both Q and R, clearly there is something
nonstandard about the situation. In this case, why should one exclude the
possibility that the members of N are evenly distributed between P and NP?
Maybe it is neither the case that most N are in P, nor that most N are in
NP. Taking the intersection of the two permitted extensions will therefore
obtain the right conclusion set (no conclusion concerning N) but for an
unconvincing reason. From the “critical review” point of view it would be
better to have a mechanism that adds both nsub(N,Q,P) and nsub(N,R,NP)
in view of the recognized anomaly. This is similar to the position taken by
traditional skeptical approaches to defeasible inheritance.
This topic may also be discussed in terms of the proportion semantics.
Consider an object domain O that is partitioned into two disjoint sets P
and NP (we assume the same encoding of the scenario as above, and identify
class names and their sets of members), assume two sets Q and R most but
not all of whose members belong to P and NP, respectively, and consider all
possible ways of choosing a set N most of whose members belong to Q ∩ R.
Question: in what percentage of the cases will the set N belong wholly or
almost wholly to P, the same for NP, and in what percentage of the cases
will neither apply? It seems safe to assume that the ’neither’ case will
dominate strongly, which means that it would be appropriate to infer both
nsub(N,Q,P) and nsub(N,R,NP). The Defeater Inheritance operation does
not do that, so in this respect it is not in line with the informal interpretation
of the Proportion Semantics.
21
One may also object against specific instances of this example, such as the
’Nixon diamond’ instance, on the grounds that the dichotomy between, for
example, “pacifist” and “non-pacifist” is too simplistic, and that a person may take intermediate positions concerning the use of military force.
However, this objection depends on the particular instance of the schema,
whereas the previously mentioned objection applies for all scenarios involving classes with a substantial number of members.
7.2
Difficulties with the Critical Review Stance
The critical review stance is attractive in principle but not so easy in realize
consistently. One problem is that because of the nature of nonmonotonic
reasoning, the addition of an abnormality literal that is made on a cautionary note may turn out to have the opposite effect of enabling a conclusion
that would otherwise not have been made. Consider for example the following inheritance network.
c sub d
d sub e
e sub g
nsx(d,e,g)
g sub q
c sub m
m sub p
p dj q
The first four literals constitute an on-path preclusion situation; the first
one plus the last four constitute a case of directly conflicting subsumers.
The use of Restriction 9 disables the side of the “Nixon diamond” that
leads from c to q, thereby enabling the conclusion c dsub p in the derived
extension. This is a cause of concern since the use of Restriction 9 serves a
cautionary purpose.
Another example of the same kind is the Cascaded Ambiguities scenario
proposed by Touretzky et al in [?], and which can be written as follows in
our representation:
N sub Q
Q sub P
N sub R
R sub NP
P dj NP
P sub AM
R sub FF
FF sub NAM
AM dj NAM
This scenario contains two structures with conflicting subsumer, one inside
the other. The Defeater Inference operation does not produce either N
sub AM nor N sub NAM, which is reasonable, but if both nsub(N,Q,P) and
nsub(N,R,NP) are imposed then one branch of the outer diamond structure
is disabled, and N sub NAM is obtained as a conclusion.
In view of these difficulties we shall proceed to study the properties of an
inference operation that minimizes nsub as an abnormality predicate. This
is done since it is in accordance with standard approaches in nonmonotonic
reasoning in general, and in spite of the reservations about whether it is the
best way to go in the long run. The use of the axioms is combined with
22
the use Restriction 9 as a heuristic and cautionary restriction. We shall
consider to what extent the effects of this inference operation are consistent
with the Proportion Semantics and the critical review position.
7.3
The Defeater Inference Operation
The Defeater Inference operation is defined as follows.
1. Consider all the valid, dsub-minimal extensions of the given inheritance network, for varying nsub.
2. Among them, select those where Restriction 9 is also satisfied.
3. Restrict that set to those members that are minimal with respect to
their contents for nsub, i.e., those for which no other member of the
set contains a strict subset of literals for nsub. These will be called
the accepted extensions.
4. Obtain the meet of all the accepted extensions. It will be called the
derived extension under the Defeater Inference operation.
Proposition 10. If N is a correct inheritance network then the derived
extension of N under the Defeater Inference operation is valid.
Proof. This follows immediately from Proposition 3.
8
Issues in Commonsense Inheritance
The research literature on defeasible multiple inheritance contains different
opinions about the proper handling of certain unusual situations, but there
is a wide agreement about what are to be considered as commonsense conclusions for some standard types of situations or schemas. In this section
we shall discuss several of these schemas from the following points of view:
• What conclusions are expected for the schema according to the existing literature
• What are the contents of the derived extension of the schema according to the Defeater Inference operation
• What ought to be the contents of the derived extension according to
the Proportion Semantics and the critical review position.
Bring in the metalevel probabilities perspective more clearly here.
8.1
Decoupling
The possibility of decoupling occurs if a class has a defeasible subsumption
(directly or indirectly) to the lowest class in an instance of the schema of
directly conflicting subsumers. For example, suppose the ‘Nixon Diamond’
example is augmented with one more class, becoming
23
N
Q
N
R
P
H
sub Q
sub P
sub R
sub NP
dj NP
sub N
Several published approaches require that there should still be two permitted extensions, namely one where both H and N are subsumed by P, and
one where they are subsumed by NP. Some however allow four extensions,
including those where N and H are assigned different subsumers. This situation is characterized as decoupling and is usually considered to be contrary
to commonsense.
It is easily verified that the Defeater Inheritance operation obtains two
of these extensions, i.e. it does not admit decoupling. This is because
nsub(N,Q,P) implies nsub(H,Q,P) according to Restriction 9, and similarly for nsub(N,R,NP).
However, in view of the earlier dicussion about directly conflicting subsumers, it is debatable whether the case of decoupling is an issue at all.
The class that may be decoupled (H in the example) is always subject to
the argument that it may be evenly split between the proposed alternatives
(P and NP in the example), just like its superior.
8.2
Special Non-decoupling Situations
In view of the strong acceptance of the schema of directly conflicting subsumers, one should anyway ask whether there are some special cases where
the schema can be defended and is supported. One obvious case is when
the class H is a singleton, but this case must be treated in the way discussed
in Section 2.5.
Another such case is when there is additional knowledge to the effect that
the members of a class tend to behave in the same way or have similar properties. Consider, for example, the situation in an electoral college consisting
of several delegations, where the members of a delegation usually vote for
the same candidate (by voluntary agreement, or due to the rules governing
the process). One can easily think of given information of the form “the
delegation from state S usually votes for the candidate from the party P”,
and there is a possibility of conflicting subsumers. If the uniform voting rule
is strict or if the number of exceptions is moderate then it can be encoded
as a proposition of the form
c sub R ∨ c sub D
(In order to allow domain knowledge to be expressed as propositions in general, it is necessary to extend the syntax so that class identifiers can occur
in propositions). It is in special cases like this that a restriction against
decoupling may be applicable.
8.3
Choice of Breakpoint
Several additional examples are shown in Appendix 2, but two scenarios
are particularly interesting and will be discussed here. Consider first the
following abstract scenario:
24
A sub B
B sub C
C sub D
D dj G
A sub G
Here the use of the Defeater Inheritance operation obtains two permitted
extensions, one containing nsub(A,B,C) and one containing nsub(A,C,D).
The extension containing nsub(B,C,D) will also contain nsub(A,C,D) by
virtue of Restriction 9, so it is not nsub-minimal, which is significant since
otherwise the conclusion B dsub D would be lost. Neither of the permitted
extensions is preferred over the other. Therefore, A dsub C is not included
in the resulting extension.
Informal application of the proportion semantics gives the same result. Because of the symmetry, the proportion semantics should not support one of
these choices over the other.
The lack of support for the conclusion A dsub C in this example is different
from what is obtained in the traditional, path-based approaches to multiple
inheritance, and anyone who is used to these approaches may consider this
to be a fault in the present approach.
We propose that it is not, however. Consider the following argument which
is a kind of defeasible reductio ad absurdum: if A were subsumed by C in a
normal way then it should have followed that A is subsumed by D, but we
know that that is not the case, therefore it is not possible to conclude that
A is subsumed by C. It is not difficult to construct scenarios that instantiate
the abstract scenario and where the counterpart of A dsub C does not hold,
for example as follows:
CitizenOfGuyana sub LivesInLatinAmerica
LivesInLatinAmerica sub SpouseHispSpeaking
SpouseHispSpeaking sub HispanicSpeaking
HispanicSpeaking dj EnglishSpeaking
CitizenOfGuyana sub EnglishSpeaking
where HispanicSpeaking is the class of those having Spanish or Portuguese
as their first language, and SpouseHispSpeaking is the class of those persons whose husband or wife has either of these as their first language. We
would not like to infer that most citizens of Guyana (which is an Englishspeaking country) are married to Hispanic-speaking persons.
The general question is where, in a chain of defeasible subsumption links,
shall the chain be broken if it is inconsistent to use the entire chain. Traditional path-based methods have not taken this issue into account.
8.4
Stein’s Floating Conclusions Scenario
The next example is due to Stein [Ste92] and was used by [MS91] as one of
their two examples of floating conclusions. It is as follows.
SeedlessGrapeVine sub GrapeVine
SeedlessGrapeVine sub SeedlessThing
SeedlessThing sub NotFruitPlant
GrapeVine sub FruitPlant
GrapeVine sub Vine
Vine sub ArborPlant
25
FruitPlant sub NotArborPlant
FruitPlant sub Tree
ArborPlant sub Plant
Tree sub Plant
(Obvious dj literals have been omitted). According to the traditional analysis, the challenge is that each extension contains a path from SeedlessGrapeVine
to Plant, but these paths are different in different extensions, and there is
no such path that occurs in all extensions.
This example illustrates several of the issues in the present topic, and it
is therefore worthwhile to study in some detail how the Defeater Inference
Operation applies to it. The reader is encouraged to follow it up by drawing
the corresponding diagrams.
Each extension must contain defeaters that avoid inconsistencies concerning whether SeedlessGrapeVine is in FruitPlant or not, and concerning
whether GrapeVine is in ArborPlant or not. Therefore, each extension
must contain one of the following two literals:
nsub(SeedlessGrapeVine,GrapeVine,FruitPlant)
nsub(SeedlessGrapeVine,SeedlessThing,NotFruitPlant)
and it must also contain one of the following:
nsub(GrapeVine,Vine,ArborPlant)
nsub(GrapeVine,FruitPlant,NotArborPlant)
For both members of the second group, Restriction 9 forces an additional
literals for nsub, namely, respectively:
nsub(SeedlessGrapeVine,Vine,ArborPlant)
nsub(SeedlessGrapeVine,FruitPlant,NotArborPlant)
This gives four combinations whose contents for nsub are as follows:
nsub(SeedlessGrapeVine,GrapeVine,FruitPlant)
nsub(GrapeVine,Vine,ArborPlant)
nsub(SeedlessGrapeVine,Vine,ArborPlant)
and
nsub(SeedlessGrapeVine,GrapeVine,FruitPlant)
nsub(GrapeVine,FruitPlant,NotArborPlant)
nsub(SeedlessGrapeVine,FruitPlant,NotArborPlant)
and
nsub(SeedlessGrapeVine,SeedlessThing,NotFruitPlant)
nsub(GrapeVine,Vine,ArborPlant)
nsub(SeedlessGrapeVine,Vine,ArborPlant)
and
nsub(SeedlessGrapeVine,SeedlessThing,NotFruitPlant)
nsub(GrapeVine,FruitPlant,NotArborPlant)
nsub(SeedlessGrapeVine,FruitPlant,NotArborPlant)
All these are consistent and obtain valid extensions. Number two and number four entail
SeedlessGrapeVine dsub ArborPlant
SeedlessGrapeVine dsub Plant
Number three entails
SeedlessGrapeVine dsub FruitPlant
SeedlessGrapeVine dsub Tree
SeedlessGrapeVine dsub Plant
26
The first combination is the most interesting one since it blocks both paths
from SeedlessGrapeVine to Plant. It is minimal with respect to contents
for nsub, like the other three, and therefore
SeedlessGrapeVine dsub Plant
is not a consequence of the given inheritance network according to the
Defeater Inference Operation. In fact, SeedlessGrapeVine is neither a
FruitPlant nor a NotFruitPlant in the first extension, and this is why
this extension does not allow the conclusion that it is a Plant.
This effect is due to Restriction 9, since if that restriction is not imposed
then the first extension can be simply
nsub(SeedlessGrapeVine,GrapeVine,FruitPlant)
nsub(GrapeVine,Vine,ArborPlant)
The case for this alternative could maybe be made informally as follows:
grapevines are fruitplants and vines; they are not arbor-plants and this is
normal for fruitplants but exceptional for vines. However, seedless grapevines
are an exceptional kind of grapevines which are not fruitplants, and therefore we take them to be more genuine vines than grapevines in general, and
in particular we take them to be arbor-plants.
This example should therefore be seen in the context of the discussion about
directly conflicting subsumers in section 7.1. If one takes the position that
every singleton and every other class that has chains of upward sub links to
both arguments of a dj literal must be subsumed by one of those arguments,
then the present inference method fails to obtain an intended conclusion.
If one takes instead a position of allowing that the class is not subsumed
by either one of the disjunction arguments, then the Defeater Inference
Operation gives the correct result in the present example. The latter view
is in line with an informal interpretation of the Proportion Semantics and
we consider it to be the correct one.
It may be argued that the very distinction between “arbor-plant” and “not
arbor-plant” implies that there can not be a third possibility. In this case
the problem is that the representation for inheritance networks that has
been introduced so far does not allow one to represent the complement of
a given class; it only allows to represent disjunction between classes. A
generalization that does allow one to represent complement classes will be
outlined in the next section. However, even so, the principle of the excluded
third can at most apply to individuals and to singleton classes; it does not
follow that it must apply to subsumed classes in general.
9
Object-level Relations and Amendments for
Description Logic
Already the earliest work on inheritance networks addressed the representation of binary relations between the objects that are members of the
classes described by the network [Tou86]. [NW01] propose the use of the
argumentation logic of [Dun95] for representation problems that involve the
combination of nonmonotonic inheritance and the use of binary relations.
Since our approach consists of a strict part and a defeasible part, using
the v and dj operators and the dsub and nsub operators, respectively,
it is straightforward to extend the strict part with additional constructs.
27
In particular, the constructs of ALC description logic [?] provide useful
additional expressivity, and they can be added easily. This extension is often
useful for representing domain-specific knowledge, and its use is illustrated
by three of the examples in the Appendix. The following technical details
are required for these examples, and could be used as the beginning of an
amalgamation of the current approach with a variety of Description Logic.
A number of binary relations R1 , R2 , ... are assumed over the object domain
O. The interpretation mapping M is extended so that for every relation
symbol or role Ri , M(Ri ) is a subset of O × O.
The syntax for inheritance networks is modified as follows. A term is an
expression that is formed in some of the following ways:
• Each class symbol is a term
• If A and B are terms, then A t B, A u B and ¬A are terms
• If A is a term and R is a role, then ∀R.A, ∃R.A and ∀∗ R.A are terms
The interpretation of these expressions is the standard one; the last mentioned term is defined as ∀R.A u ∃R.A, i.e., it is the class for all objects x
for which R(x, y) holds for at least one y, and all those y are members of A.
The definition of an inheritance network is now extended with one component so that it is a sixtuple hS, C, T, Γ, ∆, Λi S and C are like before, T is a
finite set of terms over C containing C itself as a subset, and the last three
elements are sets of literals using T instead of C.
In the definition of the proportion semantics, underlying structures are restricted to those structures where all members of T are assigned a non-empty
set of members. The assignment for a nonatomic term is chosen in the standard way as a function of the assignments to the term’s components. This
means that if T should contain a term that by definition must have an
empty set of members, then the network does not have any model. The
interpretations of the monotonic predicates v and dj are extended in the
obvious way to the case of nonatomic arguments.
Axioms 1 to 8, Restriction 9 and the Defeater Inference operation can all be
used with this extended representation. Additional axioms are also needed;
already the analysis of Stein’s Floating Conclusions Scenario showed the
need for an additional rule for terms of the form ¬A. A full treatment of
this topic is beyond the scope of the present article, but it is interesting to
note that even without the addition of further rules, this representation is
sufficient for a number of anecdotal scenario examples from earlier articles
on this topic. In particular, the cited article [NW01] contains two examples,
i.e. the “campus residence” scenario and the “good math student” scenario.
Appendix 2 shows how the present approach can represent these scenarios
and that the same conclusions are obtained and how an inconsistency in one
part of an inheritance network is kept local and does not damage inference
in other parts of the network.
We shall just make a few observations with respect to possible inference
rules. Because of the importance of the disjointness predicate in our approach, the following rule is often needed when working examples:
A dj B → ∀∗ R.A dj ∀∗ R.B
28
Since (A v B u C) follows from (A v B) and (A v C), one may be tempted
to believe that the following, similar restriction should also hold:
10.
A dsubm B ∧ A dsubn C → A dsubm+n B u C
Actually this is just almost correct, as shown by the following example
Suppose 80 percent of the members of A are members of both B and C, 10
percent are only in B, 10 percent are only in C, and the value of the constant
K is so large that chaining is admitted. The above formula would imply
that 81 percent of the members of A are in A u B, which is too strong.
A correct rule can be obtained by modifying the above rule so as to use
another operation rather than addition.
10
Alternative Approaches and Discussion
The research topic of defeasible inheritance has several synonyms, such as
multiple inheritance (with defaults), nonmonotonic inheritance, and more.
The literature on this topic is quite extensive, but with a concentration
around the 1990’s. Most work in this area has used networks with two
kinds of links, namely positive and negative, defeasible inheritance links.
Inference has usually been defined either algorithmically, or by translation
to a variety of default logic. Besides these major approaches there has been
some work in more recent years using conditional logic and argumentation
systems.
The approach that has been used in the present work represents a radical
departure from these earlier ones, both through the use of a doubt index
and through the introduction of the defeater predicate nsub. An comparison of our approach with each of the earlier ones on the logical, semantic
or algorithmic level is clearly not possible in the framework of this article.
However, we propose that it is just as important to check a new approach
for how it behaves on actual, significant examples, as to compare with earlier work on the formal level. The present section contains an admittedly
incomplete review of some earlier work, where we have favored earlier work
that do present their solutions for interesting scenario examples. Section 8
above and Appendix 2 below discuss a number of standard examples from
the point of view of our approach.
10.1
Semantics of Defeasible Inheritance
There have been a number of earlier proposals for defining the semantics
of defeasible inheritance networks, for example by Krishnaprasad et al [?],
by Doherty [?], by Schlechta [?], and by Pollock [?]. However, the purpose of these semantics is to clarify the meaning of a semantic network
and to characterize the differences between different approaches to semantic inheritance. They have not been used in conjunction with an axiomatic
characterization and for analyzing the latter. Since they do not use a doubt
index, and since they use negative arcs instead of defeater literals it would
not have been possible to use one of these definitions in conjunction with
the present work.
29
10.2
Preferential Entailment
Most work on defeasible inheritance has defined inference either by specifying a computational inference procedure, or using default logic or another,
similar set of inference rules. Work using circumscription or other, similar
ways of imposing a preference order on models include the early work of
Doherty [?] and the work of Bonatti et al [BLW06].
10.3
Priority-Ordered Links
Several authors have proposed the use of a priority ordering on links in
inheritance networks (literals, in our terminology) in order to resolve ambiguities, or for other purposes. Baader and Hollunder [BH95] used this
approach in the context of description logic. [HV02] propose a representation of nonmonotonic inheritance problems using a standard first-order
logic with the addition of a priority ordering on the restrictions. [Hor07]
uses (simple) examples of multiple inheritance to illustrate his approach to
defaults with priorities.
Applications of this kind can sometimes be re-expressed in our approach
without the use of a priority ordering and through the introduction of a
small number of additional nodes in the inheritance network. There is no
result so far concerning the range of situations where this transformation
is possible. However, the (single) example used in the article by Heymans
and Vermeir can in fact be rewritten in this way, as shown by the Juvenile
Offender example in Appendix 2.
[Mor98] introduces formula-augmented inheritance networks where nodes
in the network can have logic formulas attached to them. In her case, the
priority order is also used for resolving conflicts between inherited, attached
formulas, which may occur even though there is no path-level conflict in the
network.
10.4
Defeasible Inheritance in Description Logics
There have been a number of proposals for extending description logics
with the possibility of defeasible conclusions, beginning with the work of
Padgham and Zhang [PZ93] and of Straccia [Str93]. Other contributions
have been made e.g. by Baader and Hollunder [BH95], Donini et al [DNR02],
Rosati [Ros05], and Bonatti et al [BLW06].
A comprehensive review of these proposals is beyond the scope of the present
article. Let us however comment briefly on the last-mentioned article, where
Bonatti, Lutz and Wolter propose the use of circumscription together with
concept-based (class-based, in our terms) abnormality predicates. This proposal differs from the previous articles since those have in most cases been
based on default logic or on epistemic operators. The proposed approach is
illustrated by the following introductory example, in their notation
Mammal v ∃habitatLand t AbMammal
Whale v Mammal u ¬∃habitatLand
Minimization of AbMammal obtains the desired conclusions. This approach
necessitates the introduction of a large number of abnormality predicates.
30
In order to handle representational problems such as specificity, it allows the
definition of a priority ordering on these predicates. Usually this ordering
coincides with the subsumption hierarchy.
However, the main topic of the article is not representational issues such as
these, but the computational complexity of the proposed method.
One potential problem with the proposed approach is that there is likely to
be a need for several abnormality predicates for each class. Mammals, for
example, have a number of characteristic properties, and for many of them
there are exceptions. The representation shown above will have the effect
that a subclass that is exceptional with respect to one of these properties is
disinherited with respect to all the others as well. The natural way to solve
this would be to add a second parameter to the abnormality predicate, for
instance as follows.
Mammal v ∃habitatLand t Ab(Mammal,habitatLand)
However, the representational and computational ramifications of such a
change remain to be studied.
There is a certain resemblance between this work and ours inasmuch as both
are using preference-based approaches. However, the description-logic basis
of the work of Bonatti et al leads them to “single-argument” techniques
where information is attached to nodes in the network, both with respect to
the abnormality predicate and with respect to the separate priority ordering
for obtaining specificity. In our approach we begin by introducing predicates
of two and three arguments which merely have network nodes (classes, concepts) as some of their arguments. This is a more open representation which
has made possible both the three-argument defeater predicate, with its considerable expressivity, at the same time as the extension in the direction of
Description Logic.
10.5
Argumentation Systems
Pollock [Pol95] and Dung [Dun95] have proposed the use of argumentation
systems for defeasible reasoning, and in particular for defeasible inheritance.
For an overview of this topic, please refer to the article by Prakken and
Vreeswijk [PV02]. This approach differs strongly from other approaches to
defeasible inheritance since the others, in spite of their differences, all view
the problem as one of defining what are the conclusions from a given set of
premises. A comparison between the present work and approaches based on
argumentation systems would therefore carry too far for the present article.
The approach that we have proposed in the present article differs from
the approaches mentioned here in a number of respects, in particular, the
annotation of subsumption links with doubt values, the introduction of a
set of restrictions for specifying valid conclusions, and the use of a formal
semantics for validating the restrictions.
10.6
Reactive Diagrams
Reactive diagrams [?] is a representation system that is very general, and
which in particular claims to be able to express argumentation systems as
well as inheritance diagrams. Inference in inheritance diagrams is determined through a complex inductive algorithm on paths between pairs of
31
points in the graph. This approach is similar to ours with respect to the
use of defeater links (links that invalidate other links), but it differs in other
aspects: there is no underlying semantics, and no use of doubt indices.
11
Conclusion
The main results of this article are as follows. The use of doubt annotated
links and of defeater literals have added important expressivity to defeasible
inheritance networks. An axiomatic representation has been proposed for
defining what are valid conclusions from a given inheritance network, consisting of a set of axioms and an additional restriction which has a heuristic
and cautionary character. The proportion semantics has been defined, and
the axioms have been shown to be sound with respect to this semantics. A
possible approach to proving completeness has been described.
The Defeater Inference operation has been defined, based on the use of
the axioms, the additional restriction, and a nonmonotonic operation of
minimizing the extent of the defeater predicate. The motivation for, and
the effects of this inference operation have been discussed for a number of
important configurations of nodes in inheritance networks. The operation
obtains reasonable and intended results in most cases, but not in all of them.
The definition of an underlying semantics and a set of axioms, and the
verification of the soundness of the latter with respect to the former provides
a foundation for the analysis of a variety of nonmonotonic inference methods
for defeasible inheritance. The present article contains the beginning of such
analyses, but it also suggests a number of directions for continued work. The
following are some important topics:
• Alternative or additional heuristic restrictions, as well as modifications of the preference relation that simply minimizes nsub. There
still remain cases where the present definition does not produce reasonable results.
• Extending the representation with a modal “possibly” operator. For
scenarios with directly conflicting subsumers, such as the Nixon Diamond scenario, this would make it possible to express that both the
conflicting paths are possibly valid, which should help to avoid the
problem shown in the Cascaded Ambiguities scenario in Section 7.2.
• Combination of the present approach with Description Logic.
• Further axioms or heuristic restrictions that apply to singleton classes.
• The use of propositions for expressing domain-specific information,
and not merely for expressing axioms and heuristic restrictions.
• Robustness in the face of local inconsistencies. The solution for the
“good math student” scenario in Appendix 2 exhibits such robustness,
and it would be of interest if this is a general property of the approach.
• Properties of inheritance networks with circular structures.
• Completeness of the proposed set of axioms, or amendments to the
axioms that achieve completeness.
• Algorithms for inference in defeasible inheritance networks using the
approach proposed here.
32
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